diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 330246b..9e5fa66 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-05-20T20:12:14","documenter_version":"1.4.1"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-05-20T20:15:29","documenter_version":"1.4.1"}} \ No newline at end of file diff --git a/dev/api/index.html b/dev/api/index.html index 2bafc01..b676d59 100644 --- a/dev/api/index.html +++ b/dev/api/index.html @@ -1,2 +1,2 @@ -List of functions · Ket.jl

List of functions

Basic

Ket.ket — Function
ket([T=ComplexF64,] i::Integer, d::Integer)

Produces a ket of dimension d with nonzero element i.

source
Ket.ketbra — Function
ketbra(v::AbstractVector)

Produces a ketbra of vector v.

source
Ket.proj — Function
proj([T=ComplexF64,] i::Integer, d::Integer)

Produces a projector onto the basis state i in dimension d.

source
Ket.gell_mann! — Function
gell_mann!(res::AbstractMatrix{T}, i::Integer, j::Integer, d::Integer = 3)

In-place version of gell_mann.

source
Ket.cleanup! — Function
cleanup!(M::AbstractArray{T}; tol = Base.rtoldefault(real(T)))

Zeroes out real or imaginary parts of M that are smaller than tol.

source

Entropy

Ket.entropy — Function
entropy([b=2,] ρ::AbstractMatrix)

Computes the von Neumann entropy -tr(ρ log ρ) of a positive semidefinite operator ρ using a base b logarithm.

Reference: von Neumann entropy.

source
entropy([b=2,] p::AbstractVector)

Computes the Shannon entropy -Σᵢpᵢlog(pᵢ) of a non-negative vector p using a base b logarithm.

Reference: Entropy (information theory).

source
Ket.relative_entropy — Function
relative_entropy([b=2,] ρ::AbstractMatrix, σ::AbstractMatrix)

Computes the (quantum) relative entropy tr(ρ (log ρ - log σ)) between positive semidefinite matrices ρ and σ using a base b logarithm. Note that the support of ρ must be contained in the support of σ but for efficiency this is not checked.

Reference: Quantum relative entropy.

source
relative_entropy([b=2,] p::AbstractVector, q::AbstractVector)

Computes the relative entropy D(p||q) = Σᵢpᵢlog(pᵢ/qᵢ) between two non-negative vectors p and q using a base b logarithm. Note that the support of p must be contained in the support of q but for efficiency this is not checked.

Reference: Relative entropy.

source
Ket.binary_relative_entropy — Function
binary_relative_entropy([b=2,] p::Real, q::Real)

Computes the binary relative entropy D(p||q) = p log(p/q) + (1-p) log((1-p)/(1-q)) between two probabilities p and q using a base b logarithm.

Reference: Relative entropy.

source

Measurements

Ket.sic_povm — Function
sic_povm(d::Integer)

Constructs a vector of d² vectors |vᵢ⟩ such that |vᵢ⟩⟨vᵢ| forms a SIC-POVM of dimension d. This construction is based on the Weyl-Heisenberg fiducial.

Reference: Appleby, Yadsan-Appleby, Zauner, arXiv:1209.1813

source
Ket.test_sic — Function
test_sic(vecs)

Tests whether vecs is a vector of d² vectors |vᵢ⟩ such that |vᵢ⟩⟨vᵢ| forms a SIC-POVM of dimension d.

source
Ket.dilate_povm — Function
dilate_povm(vecs::Vector{Vector{T}})

Does the Naimark dilation of a rank-1 POVM given as a vector of vectors. This is the minimal dilation.

source
dilate_povm(E::Vector{<:AbstractMatrix})

Does the Naimark dilation of a POVM given as a vector of matrices. This always works, but is wasteful if the POVM elements are not full rank.

source
Ket.povm — Function
povm(B::Vector{<:AbstractMatrix{T}})

Creates a set of (projective) POVMs from a set of bases given as unitary matrices.

source
povm(A::Array{T, 4}, n::Vector{Int64})

Converts a set of POVMs in the common tensor format into a matrix of matrices. The second argument is fixed by the size of A but can also contain custom number of outcomes.

source
Ket.mub — Function
mub(d::Integer)

Construction of the standard complete set of MUBs. The output contains 1+minᵢ pᵢ^rᵢ bases, where d = ∏ᵢ pᵢ^rᵢ.

Reference: Durt, Englert, Bengtsson, Życzkowski, arXiv:1004.3348.

source

Nonlocality

Ket.chsh — Function
chsh([T=Float64,] d::Integer = 2)

CHSH-d nonlocal game in full probability notation. If T is an integer type the game is unnormalized.

Reference: Buhrman and Massar, arXiv:quant-ph/0409066.

source
Ket.local_bound — Function
local_bound(G::Array{T,4})

Computes the local bound of a bipartite Bell functional G, written in full probability notation as a 4-dimensional array.

source
Ket.tsirelson_bound — Function
tsirelson_bound(CG::Matrix, scenario::Vector, level::Integer)

Upper bounds the Tsirelson bound of a bipartite Bell funcional game CG, written in Collins-Gisin notation. scenario is vector detailing the number of inputs and outputs, in the order [oa, ob, ia, ib]. level is an integer determining the level of the NPA hierarchy.

This function requires Moment. It is only available if you first do "import MATLAB" or "using MATLAB".

source
Ket.correlation_tensor — Function
correlation_tensor(p::AbstractArray{T, N2}; marg::Bool = true)

Applies N sets of POVMs onto a state rho to form a probability array. Convert a 2x...x2xmx...xm probability array into

  • a mx...xm correlation array (no marginals)
  • a (m+1)x...x(m+1) correlation array (marginals).
source
Ket.probability_tensor — Function
probability_tensor(Aax::Vector{POVM{T}})

Applies N sets of POVMs onto a state rho to form a probability array.

source
Ket.fp2cg — Function
fp2cg(V::Array{T,4}) where {T <: Real}

Takes a bipartite Bell functional V in full probability notation and transforms it to Collins-Gisin notation.

source

Norms

Ket.kyfan_norm — Function
kyfan_norm(X::AbstractMatrix, k::Int, p::Real = 2)

Computes Ky-Fan (k,p) norm of matrix X.

source

Random

Ket.random_state — Function
random_state([T=ComplexF64,] d::Integer, k::Integer = d)

Produces a uniformly distributed random quantum state in dimension d with rank k.

Reference: Życzkowski and Sommers, arXiv:quant-ph/0012101.

source
Ket.random_unitary — Function
random_unitary([T=ComplexF64,] d::Integer)

Produces a Haar-random unitary matrix in dimension d. If T is a real type the output is instead a Haar-random (real) orthogonal matrix.

Reference: Mezzadri, arXiv:math-ph/0609050.

source
Ket.random_povm — Function
random_povm([T=ComplexF64,] d::Integer, n::Integer, r::Integer)

Produces a random POVM of dimension d with n outcomes and rank min(k, d).

Reference: Heinosaari et al., arXiv:1902.04751.

source

States

Ket.phiplus — Function

Produces the maximally entangled state Φ⁺ of local dimension d

source
Ket.psiminus — Function

Produces the maximally entangled state ψ⁻ of local dimension d

source
Ket.isotropic — Function

Produces the isotropic state of local dimension d with visibility v

source

Internal functions

Ket._partition — Function
partition(n::Integer, k::Integer)

If n ≥ k partitions the set 1:n into k parts as equally sized as possible. Otherwise partitions it into n parts of size 1.

source
Ket._fiducial_WH — Function
_fiducial_WH(d::Integer)

Computes the fiducial Weyl-Heisenberg vector of dimension d.

Reference: Appleby, Yadsan-Appleby, Zauner, arXiv:1209.1813 http://www.gerhardzauner.at/sicfiducials.html

source
+List of functions · Ket.jl

List of functions

Basic

Ket.ket — Function
ket([T=ComplexF64,] i::Integer, d::Integer)

Produces a ket of dimension d with nonzero element i.

source
Ket.ketbra — Function
ketbra(v::AbstractVector)

Produces a ketbra of vector v.

source
Ket.proj — Function
proj([T=ComplexF64,] i::Integer, d::Integer)

Produces a projector onto the basis state i in dimension d.

source
Ket.gell_mann! — Function
gell_mann!(res::AbstractMatrix{T}, i::Integer, j::Integer, d::Integer = 3)

In-place version of gell_mann.

source
Ket.cleanup! — Function
cleanup!(M::AbstractArray{T}; tol = Base.rtoldefault(real(T)))

Zeroes out real or imaginary parts of M that are smaller than tol.

source

Entropy

Ket.entropy — Function
entropy([b=2,] ρ::AbstractMatrix)

Computes the von Neumann entropy -tr(ρ log ρ) of a positive semidefinite operator ρ using a base b logarithm.

Reference: von Neumann entropy.

source
entropy([b=2,] p::AbstractVector)

Computes the Shannon entropy -Σᵢpᵢlog(pᵢ) of a non-negative vector p using a base b logarithm.

Reference: Entropy (information theory).

source
Ket.relative_entropy — Function
relative_entropy([b=2,] ρ::AbstractMatrix, σ::AbstractMatrix)

Computes the (quantum) relative entropy tr(ρ (log ρ - log σ)) between positive semidefinite matrices ρ and σ using a base b logarithm. Note that the support of ρ must be contained in the support of σ but for efficiency this is not checked.

Reference: Quantum relative entropy.

source
relative_entropy([b=2,] p::AbstractVector, q::AbstractVector)

Computes the relative entropy D(p||q) = Σᵢpᵢlog(pᵢ/qᵢ) between two non-negative vectors p and q using a base b logarithm. Note that the support of p must be contained in the support of q but for efficiency this is not checked.

Reference: Relative entropy.

source
Ket.binary_relative_entropy — Function
binary_relative_entropy([b=2,] p::Real, q::Real)

Computes the binary relative entropy D(p||q) = p log(p/q) + (1-p) log((1-p)/(1-q)) between two probabilities p and q using a base b logarithm.

Reference: Relative entropy.

source

Measurements

Ket.sic_povm — Function
sic_povm(d::Integer)

Constructs a vector of d² vectors |vᵢ⟩ such that |vᵢ⟩⟨vᵢ| forms a SIC-POVM of dimension d. This construction is based on the Weyl-Heisenberg fiducial.

Reference: Appleby, Yadsan-Appleby, Zauner, arXiv:1209.1813

source
Ket.test_sic — Function
test_sic(vecs)

Tests whether vecs is a vector of d² vectors |vᵢ⟩ such that |vᵢ⟩⟨vᵢ| forms a SIC-POVM of dimension d.

source
Ket.dilate_povm — Function
dilate_povm(vecs::Vector{Vector{T}})

Does the Naimark dilation of a rank-1 POVM given as a vector of vectors. This is the minimal dilation.

source
dilate_povm(E::Vector{<:AbstractMatrix})

Does the Naimark dilation of a POVM given as a vector of matrices. This always works, but is wasteful if the POVM elements are not full rank.

source
Ket.povm — Function
povm(B::Vector{<:AbstractMatrix{T}})

Creates a set of (projective) POVMs from a set of bases given as unitary matrices.

source
povm(A::Array{T, 4}, n::Vector{Int64})

Converts a set of POVMs in the common tensor format into a matrix of matrices. The second argument is fixed by the size of A but can also contain custom number of outcomes.

source
Ket.mub — Function
mub(d::Integer)

Construction of the standard complete set of MUBs. The output contains 1+minᵢ pᵢ^rᵢ bases, where d = ∏ᵢ pᵢ^rᵢ.

Reference: Durt, Englert, Bengtsson, Życzkowski, arXiv:1004.3348.

source

Nonlocality

Ket.chsh — Function
chsh([T=Float64,] d::Integer = 2)

CHSH-d nonlocal game in full probability notation. If T is an integer type the game is unnormalized.

Reference: Buhrman and Massar, arXiv:quant-ph/0409066.

source
Ket.local_bound — Function
local_bound(G::Array{T,4})

Computes the local bound of a bipartite Bell functional G, written in full probability notation as a 4-dimensional array.

source
Ket.tsirelson_bound — Function
tsirelson_bound(CG::Matrix, scenario::Vector, level::Integer)

Upper bounds the Tsirelson bound of a bipartite Bell funcional game CG, written in Collins-Gisin notation. scenario is vector detailing the number of inputs and outputs, in the order [oa, ob, ia, ib]. level is an integer determining the level of the NPA hierarchy.

This function requires Moment. It is only available if you first do "import MATLAB" or "using MATLAB".

source
Ket.correlation_tensor — Function
correlation_tensor(p::AbstractArray{T, N2}; marg::Bool = true)

Applies N sets of POVMs onto a state rho to form a probability array. Convert a 2x...x2xmx...xm probability array into

  • a mx...xm correlation array (no marginals)
  • a (m+1)x...x(m+1) correlation array (marginals).
source
Ket.probability_tensor — Function
probability_tensor(Aax::Vector{POVM{T}})

Applies N sets of POVMs onto a state rho to form a probability array.

source
Ket.fp2cg — Function
fp2cg(V::Array{T,4}) where {T <: Real}

Takes a bipartite Bell functional V in full probability notation and transforms it to Collins-Gisin notation.

source

Norms

Ket.kyfan_norm — Function
kyfan_norm(X::AbstractMatrix, k::Int, p::Real = 2)

Computes Ky-Fan (k,p) norm of matrix X.

source

Random

Ket.random_state — Function
random_state([T=ComplexF64,] d::Integer, k::Integer = d)

Produces a uniformly distributed random quantum state in dimension d with rank k.

Reference: Życzkowski and Sommers, arXiv:quant-ph/0012101.

source
Ket.random_unitary — Function
random_unitary([T=ComplexF64,] d::Integer)

Produces a Haar-random unitary matrix in dimension d. If T is a real type the output is instead a Haar-random (real) orthogonal matrix.

Reference: Mezzadri, arXiv:math-ph/0609050.

source
Ket.random_povm — Function
random_povm([T=ComplexF64,] d::Integer, n::Integer, r::Integer)

Produces a random POVM of dimension d with n outcomes and rank min(k, d).

Reference: Heinosaari et al., arXiv:1902.04751.

source

States

Ket.phiplus — Function

Produces the maximally entangled state Φ⁺ of local dimension d

source
Ket.psiminus — Function

Produces the maximally entangled state ψ⁻ of local dimension d

source
Ket.isotropic — Function

Produces the isotropic state of local dimension d with visibility v

source

Internal functions

Ket._partition — Function
partition(n::Integer, k::Integer)

If n ≥ k partitions the set 1:n into k parts as equally sized as possible. Otherwise partitions it into n parts of size 1.

source
Ket._fiducial_WH — Function
_fiducial_WH(d::Integer)

Computes the fiducial Weyl-Heisenberg vector of dimension d.

Reference: Appleby, Yadsan-Appleby, Zauner, arXiv:1209.1813 http://www.gerhardzauner.at/sicfiducials.html

source
diff --git a/dev/index.html b/dev/index.html index 0dd7019..ecbb1ff 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,2 +1,2 @@ -Home · Ket.jl

Ket.jl

Dev

Toolbox for quantum information, nonlocality, and entanglement

+Home · Ket.jl

Ket.jl

Dev

Toolbox for quantum information, nonlocality, and entanglement